Model of a quantum particle in spacetime

TL;DR

This paper reformulates a quantum particle model in spacetime, clarifying the physical meaning of internal degrees of freedom as external forces, and explores noncommuting momenta and position operators within a covariance system framework.

Contribution

It introduces a new formulation of the DFR quantum spacetime particle model, linking internal degrees of freedom to external forces and detailing the noncommutative structure of momenta and positions.

Findings

Internal degrees of freedom represent external forces.

Momentums are noncommuting and related to electromagnetic fields.

Position operators involve dual vector potentials.

Abstract

Doplicher, Fredenhagen, and Roberts (1994, 1995) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P_mu-(b/c)A_mu, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A_mu coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q_mu-(al^2/hbar c) Omega_mu where a is a generalized charge, l a fundamental length, and with vector potentials Omega_mu which are in some sense dual w.r.t. the A_mu.